| Short Courses
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| Short Courses |
1. Differentially Flat Systems
This lecture introduces the flatness property for non linear differential systems and explains how to
identify and apply this concept in the design of effective control systems.
Contents (3 hours):
- Introduction to non linear systems
- Definition of differential flatness
- Examples
- Properties of differential flat systems
- Non linear control of differential flat systems
o Application to underactuated systems
o Application to trajectory tracking
o Robustness issues
- Fault detection and identification for differential flat systems
- Applications in aeronautics
- Perspectives
Bibliographie:
Nonlinear Systems by Hassan K. Hhalil, Ptrentice hall, 2002;
Differentially Flat Systems (Control Engineering) by Hebertt Sira-Ramírez and Sunil K. Agrawal , CRC, 2004;
Flatness and defect of non-linear systems: theory and examples, M.Fliess, J.Lévine, P.Martin and P.Rouchon, International Journal of Control, vol 61, n°6, pp.1327-1361;
Nonlinear control structures and rotorcraft positioning, A. Drouin, T. Miquel and F. Mora-Camino, AIAA-Guidance Navigation and Control, Honolulu, 2008;
Rotorcraft Fault Detection Using Difference Flatness, Nan Zhang, Juan Luis López Fernandez, Andrei Doncescu and Felix Mora-Camino, AIAA Guidance Navigation and Control, Chicago 2009;
Organizers:
Prof.Dr. Félix Mora-Camino COPPE, UFRJ
Félix Mora-Camino is the head of the Automation Laboratory at ENAC ( the french civil aviation institute)
in Toulouse where he teaches automatic control and avionics systems design. Once Mr Mora-Camino was a titular
professor at COPPE/UFRJ and then a visiting professor at the Royal Air Force Academy of Marrocco in Marrakech
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2. Modern Multiparameter Stability Theory and Applications in Mechanics.
Alexander P. Seyranian
Institute of Mechanics
Moscow State Lomonosov University
A new multiparameter bifurcation theory of eigenvalues of matrix and linear differential operators is presented which is a key point for stability study of systems with finite degrees of freedom as well as distributed systems. Two important cases of strong and weak interactions (collisions) are distinguished and geometrical interpretation of these interactions is given. The presence of several parameters and the absence of differentiability of multiple eigenvalues constitute the main mathematical difficulty of the analysis. With presented multiparameter bifurcation theory of eigenvalues we analyze singularities of stability boundaries and give a consistent description and explanation for several interesting mechanical effects like gyroscopic stabilization, flutter and divergence instabilities, transference of instability between eigenvalue branches, destabilization and stabilization by small damping, disappearance of flutter instability, parametric resonance in periodically excited systems etc.
A significant part of the work is devoted to difficult stability problems of periodic systems dependent on multiple constant parameters. This subject has been a challenge for more than one hundred years since Mathieu, Floquet, Hill, Rayleigh, Lyapunov, Poincare. From the very beginning these problems were multiparameter. In the present work, with the bifurcation theory of multipliers, geometrical description of the stability boundary and its singularities for periodic systems is given. Then we formulate and solve parametric resonance problems for one- and multiple degrees of freedom systems in three-dimensional space of physical parameters: excitation frequency, amplitude, and viscous damping coefficient. The main result obtained here is that we find the instability (parametric and combination resonance) domains as half-cones in three-parameter space with the use of eigenfrequencies and eigenmodes of the corresponding conservative system. The experiments on parametric resonance of pendulums and elastic beams confirm validity and accuracy of the obtained theoretical results.
The swing problem is undoubtedly among the classical problems of mechanics. It is known from practice that to set a swing into motion one should erect when the swing is in limit positions and squat when it is in the middle vertical position, i.e. carry out oscillations with double the natural frequency of the swing. However in the literature you can not find formulae for instability regions explaining the phenomenon of swinging. In the present paper the simplest model of the swing is described by a massless rod with a concentrated mass periodically sliding along the rod axis. Based on analysis of multipliers the asymptotic formulae for instability (parametric resonance) domains in three-dimensional parameter space are derived and analyzed.
Another classical problem is the problem of finding instability regions for a system with periodically varying moment of inertia. An equation describing small torsional oscillations of the system with periodic coefficients dependent on four parameters including damping is derived. Analytical results for instability (parametric resonance) regions in parameter space are obtained and numerical examples are presented.
A problem of stabilization of a vertical (inverted) position of a pendulum under high frequency vibration of the suspension point is considered. Small viscous damping is taken into account, and periodic excitation function describing vibration of the suspension point is assumed to be arbitrary. A formula for stability region of Hill’s equation with damping near zero frequency is obtained. For several examples it is shown that analytical and numerical results are in a good agreement with each other. An asymptotic formula for stabilization region of the inverted pendulum with arbitrary periodic excitation function is derived. It is shown that the effect of small viscous damping is of the third order, and taking it into account leads to increasing critical stabilization frequency.
References:
[1] Seyranian, A. P., Mailybaev, A.A. 2004. Multiparameter Stability Theory with Mechanical Applications, New Jersey: World Scientific.
[2] Seyranian, A.P. Parametric resonance of a swing. J. Appl. Maths Mechs. 2004.Vol. 68. ? 5. P. 338-342.
[3] Seyranian, A.P. Kirillov, O.N., Mailybaev A.A. Coupling of eigenvalues of complex matrices at diabolic and exceptional points. Journal of Physics A: Mathematical and General. 2005.Vol. 38. No. 8. P. 1723-1740.
[4] Kirillov O.N., Seyranian A.P. The effect of small internal and external damping on the stability of distributed non-conservative systems. J. Appl. Math. Mech. 2005. Vol. 69. No. 4. P. 529-552.
[5] Cattani, C., Seyranian, A.P. The regions of instability of a system with a periodically varying moment of inertia. J. Appl. Math. Mech. 2005, Vol. 69 (6), pp. 810-815.
[6] Seyranian, A.?., Seyranian, A.P. The stability of an inverted pendulum with a vibrating suspension point. J. Appl. Math. Mech. 2006. Vol. 70. P. 754-761.
[7] Mailybaev, A.A., Seyranian, A.P. Bifurcations of equilibria in potential systems at bimodal critical points. Trans.ASME. Journal of Applied Mechanics, 2008, Vol. 75, P. 021016-1 - 021016-11.
[8] Seyranian, A.P., Belyakov, A.O. Swing dynamics. Doklady Physics, 2008, Vol. 53, No. 7, P. 388-394.
[9] Seyranian, A.?., Seyranian, A.P. Chelomei’s problem of the stabilization of a statically unstable rod by means of vibration. J. Appl. Math. Mech. 2008. Vol. 72. P. 898-903.
[10] Belyakov, A.O., Seyranian, A.P., Luongo A. Dynamics of the pendulum with periodically varying length. Physica D. 2009. Vol. 238. P. 1589-1597.
[11] Mailybaev, A.A., Seyranian, A.P. Stabilization of statically unstable systems by parametric excitation. Journal of Sound and Vibration. 2009. Vol. 323. P. 1016–1031.
Organizer:
Alexander P. Seyranian
Institute of Mechanics,
Moscow State Lomonosov University,
Michurinsky pr. 1, 119192 Moscow, RUSSIA
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